The present invention relates to a control device for a wide speed range induction machine, carrying out a control based upon a flux estimation performed by means of a Luenberger state observer.
The present invention moreover relates to a method for positioning the eigenvalues of a Luenberger state observer implemented in a control device for a wide speed range induction machine in order to perform an estimation of the flux of said induction machine.
As is known, direct field-orientation control of induction machines calls for the knowledge of the components of the rotor flux of the induction machines themselves, which can either be measured directly using magnetic sensors or can be estimated on the basis of the values of the quantities present at the machine terminals.
In practical applications, however, recourse is almost exclusively had to the estimation of the rotor flux of the asynchronous machine, using an appropriate mathematical model, in so far as direct measurement presents numerous problems of a technical and economic nature.
One of the most interesting solutions for the estimation of the rotor flux of induction machines is represented by the known Luenberger state observer, which is based upon the dynamic equations of the induction machine itself and upon the measurement of speed in order to reconstruct the state of the induction machine in terms of stator current and rotor flux.
FIG. 1 shows, purely by way of non-limiting example, a general diagram of the circuit of the control of an induction machine 1, represented for reasons of simplicity by the symbol of a three-phase electric motor, said control being implemented on the basis of the flux estimation performed by means of a Luenberger state observer.
As is shown in FIG. 1, the control of the induction machine 1 envisages the use of a power driving device 2 which supplies to the induction machine 1 three control voltages, the so-called three phases of a three-phase system, and is controlled by a control device 4, typically implemented by means of a digital signal processor, commonly referred to as DSP.
In particular, the control device 4 carries out control of the induction machine 1 on the basis of the estimation of the rotor flux of the induction machine 1 performed by means of the aforesaid Luenberger state observer, and receives at input one or more reference signals, among which a reference speed signal indicating the desired speed of rotation of the induction machine 1, and a plurality of measurement signals indicating the values of quantities of the induction machine 1, among which the value of the effective speed of rotation of the induction machine 1 measured by means of a speed sensor (not shown) connected to the induction machine 1 and the values of two of the three phase currents absorbed by the induction machine 1 and measured by means of appropriate sensors (not shown) at the terminals of the induction machine 1.
FIG. 2 shows the block diagram of a part of the system with which the induction machine 1 and the Luenberger state observer used in the control device 4 are conventionally modelled from the control standpoint.
As shown in FIG. 2, the induction machine is schematically represented by a block 1 defined by a state vector x formed by the state variables of the induction machine 1, of which only one part is physically measured. The block 1 receives at input an input vector u representing input quantities supplied to the induction machine 1 (control voltages supplied by the power driving device) and generates at output a measured output vector y representing measured output quantities supplied to the induction machine 1 (absorbed currents), which, as is known, are selected from among the state variables contained in the state vector x.
As is known, then, the induction machine 1 is defined by the following variable-parameter linear differential state equations:                     {                                                                              x                  .                                =                                                                            A                      ⁡                                              (                        ω                        )                                                              ⁢                    x                                    +                  Bu                                                                                                        y                =                Cx                                                                        (        1        )            
where:
A is known in the literature by the name of matrix of the induction machine or matrix of free evolution;
B is known in the literature by the name of input matrix;
and
C is known in the literature by the name of output matrix, which enables generation of the measured output vector y starting from the state vector x, i.e., selection, from among all the state variables, of the output variables of the system.
The Luenberger state observer implemented by the control device 4 is instead schematically represented by a block 6 which receives at input the input vector u and supplies at output an estimated output vector {tilde over (y)} representing the output quantities estimated by the Luenberger state observer 6, and an estimated state vector {tilde over (x)} representing the state variables estimated by the Luenberger state observer 6.
The measured output vector y and the estimated output vector {tilde over (y)} are then supplied at input to a block 8 which performs the subtraction and generates at output an error vector e supplied at input to the block 10 to carry out feedback.
As is known, then, the measured output vector y, together with the aforementioned reference signals, is supplied at input to a regulator block (not shown), which generates the aforesaid input vector u for regulating the currents and flux of the induction machine 1, in order to obtain the references required.
The Luenberger state observer applied to the dynamic model of the induction machine expressed as a function of the rotor flux and of the stator current (this being the most convenient form in so far as one of the state variables coincides with the output itself of the system, and the structure of the algorithm is thus simplified), represented on the stationary xcex1xcex2 reference system, is described by the following variable-parameter linear differential state equations and in complex form (the variables marked with the tilde refer to the variables estimated by the Luenberger state observer):                     {                                                                                                  x                    .                                    ~                                =                                                                            A                      ⁡                                              (                        ω                        )                                                              ⁢                                          x                      ~                                                        +                  Bu                  -                                      K                    ⁡                                          (                                              y                        -                                                  y                          ~                                                                    )                                                                                                                                                                y                  ~                                =                                  C                  ⁢                                      xe2x80x83                                    ⁢                                      x                    ~                                                                                                          (        2        )                                          A          =                      [                                                                                                      -                                              1                        σ                                                              ⁢                                          (                                                                                                    R                            s                                                                                L                            s                                                                          +                                                                                                            (                                                              1                                -                                σ                                                            )                                                        ⁢                                                          R                              r                                                                                                            L                            r                                                                                              )                                                                                                                                                          1                        -                        σ                                                                    σ                        ⁢                                                  xe2x80x83                                                ⁢                                                  L                          m                                                                                      ⁢                                          (                                                                                                    R                            r                                                                                L                            r                                                                          -                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                                                                    )                                                                                                                                                              L                      m                                        ⁢                                                                  R                        r                                                                    L                        r                                                                                                                                  (                                                                  -                                                                              R                            r                                                                                L                            r                                                                                              +                                              j                        ⁢                                                  xe2x80x83                                                ⁢                        ω                                                              )                                                                        ]                          ⁢                  
                ⁢                  B          =                                                    [                                                                                                    1                                                  σ                          ⁢                                                      xe2x80x83                                                    ⁢                                                      L                            s                                                                                                                                                                          0                                                                      ]                            ⁢                              xe2x80x83                            ⁢              C                        =                          [                              1                ⁢                                  xe2x80x83                                ⁢                0                            ]                                                          (        3        )                                          x          ~                =                              [                                                                                                      i                      ~                                        s                                                                                                                                          λ                      ~                                        r                                                                        ]                    =                                    [                                                                                                                                            i                          ~                                                                          s                          ⁢                                                      xe2x80x83                                                    ⁢                          α                                                                    +                                              j                        ⁢                                                  xe2x80x83                                                ⁢                                                                              i                            ~                                                                                s                            ⁢                                                          xe2x80x83                                                        ⁢                            β                                                                                                                                                                                                                                                            λ                          ~                                                ra                                            +                                              j                        ⁢                                                  xe2x80x83                                                ⁢                                                                              λ                            ~                                                                                r                            ⁢                                                          xe2x80x83                                                        ⁢                            β                                                                                                                                                          ]                        ⁢                          xe2x80x83                        ⁢                                                                                u                    =                                                                  v                        s                                            =                                                                        v                                                      s                            ⁢                                                          xe2x80x83                                                        ⁢                            α                                                                          +                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                                                      v                                                          s                              ⁢                                                              xe2x80x83                                                            ⁢                              β                                                                                                                                                                                                                                            y                    =                                                                  i                        s                                            =                                                                        i                                                      s                            ⁢                                                          xe2x80x83                                                        ⁢                            α                                                                          +                                                  ji                                                      s                            ⁢                                                          xe2x80x83                                                        ⁢                            β                                                                                                                                                                                                      (        4        )                                K        =                              [                                                                                                      k                      11                                        +                                          jk                      12                                                                                                                                                              k                      21                                        +                                          jk                      22                                                                                            ]                    =                      [                                                                                k                    1                                                                                                                    k                    2                                                                        ]                                              (        5        )            
where:
A, B and C have the meanings specified previously;
K is the feedback matrix of the Luenberger state observer;
Rs is the stator resistance of the induction machine;
Rr is the rotor resistance of the induction machine;
Lm is the magnetization inductance of the induction machine;
Ls is the stator inductance of the induction machine;
Lr is the rotor inductance of the induction machine;
"sgr" is the coefficient of dispersion of the induction machine defined as follows:   σ  =      1    -                  L        m        2                              L          s                ⁢                  L          r                    
xcfx89 is the equivalent mechanical speed of rotation of the induction machine, corresponding to the mechanical speed of rotation multiplied by the pole torques;
is xcex1,xcex2 are the components of the stator current in the stationary xcex1xcex2 reference system;
xcexr xcex1,xcex2 are the components of the rotor flux in the stationary xcex1xcex2 reference system; and
xcexds xcex1,xcex2 are the components of the stator voltage in the stationary xcex1xcex2 reference system.
Note that in equations (2) and (3) there appear the matrices A, B and C of the induction machine in so far as it is assumed that the parameters of the induction machine are known, and hence, in particular, that the matrix of the Luenberger state observer coincides with the matrix A of the induction machine.
To simplify the notation we put:                               A          =                                    [                                                                                          a                      1                                                                                                  -                                              β                        ⁡                                                  (                                                                                    a                              2                                                        +                                                          j                              ⁢                                                              xe2x80x83                                                            ⁢                              ω                                                                                )                                                                                                                                                                                a                      3                                                                                                  (                                                                        a                          2                                                +                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                                                                    )                                                                                  ]                        =                          [                                                                                          a                      1                                                                                                                          -                        β                                            ⁢                                              xe2x80x83                                            ⁢                      w                                                                                                                                  a                      3                                                                            w                                                              ]                                      ⁢                  xe2x80x83                ⁢                  
                ⁢                  B          =                      [                                                                                b                    1                                                                                                0                                                      ]                                              (        6        )            
the coefficients introduced having been defined as follows:                                           a            1                    =                                                    -                                  1                  σ                                            ⁢                              (                                                      1                                          T                      s                                                        +                                                            1                      -                      σ                                                              T                      r                                                                      )                            ⁢                              xe2x80x83                            ⁢                              a                2                                      =                                                            -                                      1                                          T                      r                                                                      ⁢                                  xe2x80x83                                ⁢                                  a                  3                                            =                              -                                                      L                    m                                                        T                    r                                                                                      ⁢                  
                ⁢                  β          =                                                                      1                  -                  σ                                                  σ                  ⁢                                      xe2x80x83                                    ⁢                                      L                    m                                                              ⁢                              xe2x80x83                            ⁢              w                        =                                          a                2                            +                              j                ⁢                                  xe2x80x83                                ⁢                ω                                                                        (        7        )            
Positioning of the eigenvalues of the Luenberger state observer, which are commonly referred to in the control field also by the term xe2x80x98polesxe2x80x99 (this term, however, is not altogether correct in so far as the system is time variant) is via the aforesaid feedback matrix K which multiplies the error vector given by the difference between the output vector measured and the output vector estimated by the Luenberger state observer.
In particular, typically each of the eigenvalues of the continuous-time Luenberger state observer is rendered proportional to a corresponding pole of the induction machine via the technique proposed in:
xe2x80x9cNew Adaptive Flux Observer of Induction Motor for Wide Speed of rotation Range Motor Drivesxe2x80x9d, Kubota, H., Matsuse, K., Nakano, T., Industrial Electronics Society, 1990. IECON ""90., 16th Annual Conference of IEEE, 1990, pp. 921-926 Vol. 2; and in
xe2x80x9cDSP-based Speed Adaptive Flux Observer of Induction Motorxe2x80x9d, Kubota, H., Matsuse, K., Nakano, T., Industry Applications, IEEE Transactions, Vol. 29, 2 March-April 1993 pp. 344-348.
This technique enables real-time updating of the coefficients of the feedback matrix K with simple algebraic operations.
In detail, the aforesaid technique envisages that, in the conventional representation of the poles and eigenvalues in magnitude and phase, the magnitudes of the eigenvalues of the continuous-time Luenberger state observer are rendered proportional to the magnitudes of the corresponding poles of the induction machine, and that the phases of the eigenvalues of the continuous-time Luenberger state observer remain unvaried with respect to those of the corresponding poles of the induction machine, i.e., in other words, they are exactly equal to the phases of the corresponding poles of the induction machine.
If, instead, the conventional partly real and partly imaginary representation is considered, the aforesaid technique envisages that the real parts and the imaginary parts of the eigenvalues of the continuous-time Luenberger state observer are rendered proportional to the real parts and, respectively, to the imaginary parts of the corresponding poles of the induction machine.
According to said technique, moreover, with the assumption made above, the matrix Ao known in the literature by the name of matrix of the estimation error of the Luenberger state observer (often, for reasons of simplicity, also referred to, albeit somewhat improperly, as matrix of the Luenberger state observer), is:                               A          o                =                              A            -            KC                    =                                                    [                                                                                                    a                        1                                                                                                                                      -                          β                                                ⁢                                                  xe2x80x83                                                ⁢                        w                                                                                                                                                a                        3                                                                                    w                                                                      ]                            -                                                [                                                                                                              k                          1                                                                                                                                                              k                          2                                                                                                      ]                                ⁢                                  xe2x80x83                                [                                  1                  ⁢                                      xe2x80x83                                    ⁢                  0                                ]                                      =                          [                                                                                          (                                                                        a                          1                                                -                                                  k                          1                                                                    )                                                                                                                          -                        β                                            ⁢                                              xe2x80x83                                            ⁢                      w                                                                                                                                  (                                                                        a                          3                                                -                                                  k                          2                                                                    )                                                                            w                                                              ]                                                          (        8        )            
whence the terms of the so-called characteristic polynomial of the Luenberger state observer may be immediately obtained as follows:                     {                                                                                                  p                                          1                      ⁢                      o                                                        +                                      p                                          2                      ⁢                      o                                                                      =                                                      a                    1                                    -                                      k                    1                                    +                                      j                    ⁢                                          xe2x80x83                                        ⁢                    w                                                                                                                                                                p                                          1                      ⁢                      o                                                        ⁢                                      p                                          2                      ⁢                      o                                                                      =                                  w                  ⁡                                      [                                                                  (                                                                              a                            1                                                    -                                                      k                            1                                                                          )                                            +                                              β                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    a                              3                                                        -                                                          k                              2                                                                                )                                                                                      ]                                                                                                          (        10        )            
and hence the analytical expression of the feedback matrix K, which makes it possible to render the eigenvalues of the matrix Ao, often referred to, for reasons of simplicity, by the term eigenvalues of the Luenberger state observer, proportional to the poles of the induction machine, as a function of the machine parameters and of the speed:                     {                                                                                                                        k                      11                                        =                                                                  (                                                  1                          -                          α                                                )                                            ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              a                            1                                                    +                                                      a                            2                                                                          )                                                                                                                                                                                    k                      12                                        =                                                                  (                                                  1                          -                          α                                                )                                            ⁢                      ω                                                                                                                                                              k                      21                                        =                                                                                            (                                                      1                            -                                                          α                              2                                                                                )                                                ⁢                                                  (                                                                                    α                              3                                                        +                                                          γ                              ⁢                                                              xe2x80x83                                                            ⁢                                                              a                                1                                                                                                              )                                                                    -                                              γ                        ⁢                                                  xe2x80x83                                                ⁢                                                  k                          11                                                                                                                                                                                                            k                      22                                        =                                                                  -                        γ                                            ⁢                                              xe2x80x83                                            ⁢                                              k                        12                                                                                                                  ⁢                          xe2x80x83                        ⁢            γ                    =                                    1              β                        =                                          σ                ⁢                                  xe2x80x83                                ⁢                                  L                  m                                                            1                -                σ                                                                        (        11        )            
The coefficient xcex1 in equation (11) expresses the factor of proportionality between the eigenvalues of the Luenberger state observer and the poles of the induction machine and makes it possible to position the eigenvalues of the Luenberger state observer with one degree of freedom.
FIG. 3 shows the trajectories of the poles of the induction machine and of the eigenvalues of the Luenberger state observer on the plane of the Laplace variable s as a function of the speed of rotation of the induction machine, for a value of xcex1 equal to 2.
It may be noted that for each value of the speed of rotation the eigenvalues of the Luenberger state observer are displaced along radial directions, and the entire trajectory or path is amplified by a scale factor, modifying the natural frequency of the eigenvalues but not their damping.
The values given in the graph of FIG. 3 refer to a three-phase electric motor characterized by the following parameters:
Discrete-time implementation of the Luenberger state observer on digital signal processors requires the calculation of the exponential matrix of a variable-parameter fourth-order system, which cannot be performed in real time in an exact way.
Generally, on-line updating of the model of the Luenberger state observer is preferred, using approximate formulas of the exponential matrix which are obtained by truncating the Taylor series at the first or second term.
The algorithms that use the first term of the Taylor series require a minimal computational effort but prove unstable typically beyond 4000 r.p.m. In addition, the approximations introduced in the implementation of the Luenberger state observer in digital form condition the precision achievable in the estimation of the state variables; consequently, in order to enable stable and proper operation over a wide range of speeds it is necessary to use second-order models, the implementation of which, however, entails a computational effort which is approximately four times greater.
Other solutions proposed envisage, instead, an interpolation between various discrete-time models calculated for certain speed values and stored in a memory table, but these techniques involve a computational effort approximately twice as great, which limits the sampling frequency and the maximum speed of operation of the induction machine.
Various other techniques for positioning the eigenvalues of the Luenberger state observer, different from the one described above in which the eigenvalues of the continuous-time Luenberger state observer are rendered proportional to those of the induction machine, have then been proposed to extend the range of stability of the Luenberger state observer. All these techniques, however, give rise to feedback matrices K that in general have somewhat complex analytical formulations and comprise operations of raising to a power, operations of division, trigonometric relations, etc., the execution of which always requires a considerable computational effort.
The aim of the present invention is therefore to provide a method for positioning the eigenvalues of the Luenberger state observer and to implement a control device for an induction machine that carries out a control based upon the flux estimation performed by means of the Luenberger state observer, said method and said control device making it possible to extend the range of stability of the Luenberger state observer considerably, using just the first-order approximation of the Taylor series of the exponential matrix, so involving minimal computational effort and thus enabling implementation thereof on processors having contained performance and cost.
According to the present invention there is provided a control device for a wide speed range induction machine, carrying out a control based upon a flux estimation performed by means of a Luenberger state observer, as defined in claim 1.
Moreover, according to the present invention there is provided a method for positioning the eigenvalues of a Luenberger state observer implemented in a control device for a wide speed range induction machine to perform an estimation of the flux of said induction machine, as defined in claim 7.